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Nonlinear characteristics of the surface air temperature over Canada Wu, Aiming; Hsieh, William W.; Shabbar, Amir 2002

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Nonlinear characteristics of the surface air temperature over CanadaAiming Wu and William W. HsiehDepartment of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, CanadaAmir ShabbarClimate Research Branch, Meteorological Service of Canada, Downsview, Ontario, CanadaReceived 10 July 2001; revised 12 May 2002; accepted 12 June 2002; published 8 November 2002.[1] Nonlinear characteristics of the Canadian surface air temperature (SAT) wereinvestigated by applying a neural-network-based nonlinear principal component analysis(NLPCA) method to the SAT anomaly data for individual seasons. The SAT data wereseparated into three subsets: data for 1900–1949 and 1900–1995 over southern Canada(south of 60C176N), called S0049 and S0095, respectively, and data for 1950–1995 over theentire country, called C5095. The NLPCA was computed for the three data setsseparately. The leading NLPCA modes from C5095 and S0095 show similar results: thenonlinearity is strong in winter (December, January, and February, DJF) and fall(September, October, and November, SON), but is much weaker in spring (March, Apriland May, MAM) and summer (June, July, and August, JJA), manifesting the seasonaldependence of the nonlinearity in the Canadian SAT. No significant nonlinearity isdetected from data set S0049, even for the winter and fall seasons, indicating interdecadaldependence of the nonlinearity. The leading NLPCA mode combines the effects ofPacific-North America (PNA) pattern and North Atlantic Oscillation (NAO) on theCanadian winter SAT. A possible reason for the existence of nonlinearity in the winterSAT only after 1950 is that the NAO manifested its strong negative phase from the 1950sto the early 1970s. INDEX TERMS: 3309 Meteorology and Atmospheric Dynamics: Climatology(1620); 3220 Mathematical Geophysics: Nonlinear dynamics; 4215 Oceanography: General: Climate andinterannual variability (3309); KEYWORDS: surface air temperature, Canada, neural network, nonlinearprincipal component analysis (NCPCA), interdecadalCitation: Wu, A., W. W. Hsieh, and A. Shabbar, Nonlinear characteristics of the surface air temperature over Canada, J. Geophys.Res., 107(D21), 4571, doi:10.1029/2001JD001090, 2002.1. Introduction[2] North America is an area of interest for climatestudies related to the El Nin˜o-Southern Oscillation (ENSO)phenomenon [Horel and Wallace, 1981; Wallace and Gut-zler, 1981; Trenberth et al., 1998]. The traditional view isthat the climate variations associated with El Nin˜o-SouthernOscillation (ENSO) are linear, with anomalies during the ElNin˜o phase being the reverse of those during the La Nin˜aphase [e.g., Ropelewski and Halpert, 1989; Bunkers et al.,1996]. However, recent evidence shows that the atmos-pheric responses to warm and cold events were not exactlyopposite. Richman and Montroy [1996] examined the com-posite January temperature and precipitation patterns overthe United States and parts of Canada associated with ElNin˜o and La Nin˜a events. Their results suggest that El Nin˜oand La Nin˜a have their own unique characteristics in termsof temperature and precipitation, so the responses are notlinear. Asymmetric spatial patterns of the Canadian surfaceair temperature (SAT) and precipitation associated with theSouthern Oscillation (SO) were detected by Shabbar andKhandekar [1996] and Shabbar et al. [1997]. Furtherevidences of nonlinear response of North America climateto ENSO were provided by Hoerling et al. [1997], whosuggested that the midlatitude atmospheric response to thedifferent phases of the SO is inherently nonlinear, due todifferences in the locations of the intense tropical PacificSST-induced deep convection between El Nin˜o and La Nin˜aevents. From the phase shift in the midlatitude geopotentialheight anomalies during the opposite phases of the SO, theyconcluded that midlatitude temperature and precipitationpatterns should also have nonlinear relations with the SO.The robustness of nonlinear climate response to ENSO’sextreme phases was then confirmed by four GCMs [Hoerl-ing et al., 2001]. A nonlinear identification of the atmos-pheric response to ENSO was also addressed by Hannachi[2001] using state-of-the-art general circulation models.[3] Describing the nonlinear behavior in the midlatitudeclimate and its nonlinear relations to ENSO is a greatchallenge. Standard multivariate statistical techniques suchas principal component analysis (PCA, also known as EOFanalysis) and canonical correlation analysis (CCA) are linearmethods. Composite analysis does not assume linearity, butJOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. D21, 4571, doi:10.1029/2001JD001090, 2002Copyright 2002 by the American Geophysical Union.0148-0227/02/2001JD001090$09.00ACL 8 - 1is restricted to the analysis of differences between specificphases of the SO or equatorial sea surface temperature (SST)indices. Are there general methods which can describe thenonlinearity of the midlatitude climate variability and itsnonlinear relations to ENSO? Recently, neural networks(NN) [Hsieh and Tang, 1998] have been used for nonlinearPCA (NLPCA) [Kramer, 1991] and nonlinear CCA(NLCCA) [Hsieh, 2000]. The tropical Pacific SST and sealevel pressure fields have recently been analyzed by theNLPCA [Monahan, 2001] and by NLCCA [Hsieh, 2001a],where SST was found to exhibit considerable nonlinearity,while the sea level pressure was found to be less nonlinear.Also, NLPCA was applied to the analysis of the winterNorthern hemisphere atmospheric variability [Monahan etal., 2000]. Hsieh [2001b] pointed out that NLPCA unifiesthe PCA and rotated PCA approaches.[4] In this paper, the NLPCA model of Hsieh [2001b] willbe applied to study the SAT variability over Canada. Themotivation of this paper is to examine the nonlinear charac-teristics of the Canadian SATusing NLPCA, before we buildour prediction models using NLCCA to link the SATand thetropical Pacific SST. The paper is organized as follows: Thedata are briefly introduced in section 2. The leading NLPCAmodes for the Canadian SAT data for 1950–1995 over theentire country, and for 1900–1995 and 1900–1949 oversouthern Canada (south of 60C176N) are presented in section 3.Some possible dynamics related to the NLPCA mode ofCanadian winter SATare discussed in section 4. A Summarywith concluding remarks is given in section 5. Details of theNLPCA model are described in Appendix A.2. Data[5] The basic data we used in this study is the griddedmonthly mean SATs interpolated from the station observa-tions for the period of January 1900 to December 1995[Vincent and Gullet, 1999]. With this data (up to 1998),Zhang et al. [2000] analyzed the trend of Canadian SAT inthe 20th century. Following Zhang et al. [2000], weseparated the data into three subsets: data for 1900–1949and for 1900–1995 over southern Canada (south of 60C176N),called S0049 and S0095, respectively, and data for 1950–1995 over whole Canada, called C5095. There are tworeasons for doing so: (1) the limited data availability innorthern Canada prior to 1950; (2) a check on the signifi-cance of the NLPCA results. If the NLPCA results arereproducible from different data sets, then the nonlinearitycan be regarded as robust rather than sampling dependent.The SAT data have an approximately linear trend [seeZhang et al., 2000, Figure 3], so linear detrending wasperformed first. Monthly SAT anomalies were calculated byremoving the climatological monthly mean based on thewhole period of each data set. The anomaly data were thensmoothed with a 3-month running mean. The SAT anoma-lies were separated into four seasons: SAT anomalies inDecember, January, and February (DJF) were used to form adata set for the winter season, March, April, and May(MAM) for the spring, June, July, and August (JJA) forthe summer, and September, October, and November (SON)for the fall.[6] The 500-mb geopotential height came from theNational Centers for Environmental Prediction’s (NCEP)reanalysis data sets for the period from January 1948 toDecember 1995 with a 2.5C176 grid over a global domain[Kalnay et al., 1996]. The monthly SST was from thereconstructed global historical SST data sets by Smith etal. [1996] for the period 1950–2000 with a resolution of 2C176by 2C176 over global oceans. Anomalies for the 500-mb heightand SST fields are calculated with respect to climatologicalmonthly means for 1950–1995. Linear detrending and 3-month running mean were then performed on both data sets.[7] The winter (December through March) index of theNorth Atlantic Oscillation (NAO), defined as the differenceof normalized sea level pressure (SLP) between Lisbon,Portugal and Stykkisholmur/Reykjavik, Iceland since 1864to 2001, was provided by Dr. Jim Hurrel of the NationalCenter for Atmospheric Research (NCAR). The SLPanomalies at each station were normalized by dividing theseasonal mean pressure by the long-term (1864–1983)standard deviation. The NAO index used in this paper isan update of the time series published by Hurrel [1995].3. The Leading NLPCA Mode of Canadian SAT3.1. C5095[8] Prior to NLPCA, traditional PCA (or EOF analysis)was performed on the SAT anomalies of each season.Variance contributions from the four leading modes arelisted in Table 1 and the spatial patterns for the three leadingmodes are shown in Figure 1. We can see that spatialTable 1. Variance Explained by the Four Leading EOF (PCA) Modes of the Seasonal SAT AnomaliesaC5095 S0095 S0049DJF MAM JJA SON DJF MAM JJA SON DJF MAM JJA SON1 44.6 46.1 40.6 40.1 58.2 54.9 48.3 55.4 61.4 55.3 47.9 60.62 24.7 17.5 15.3 29.4 21.3 21.3 16.5 24.9 19.0 20.9 15.7 20.93 11.0 14.2 14.3 10.6 7.5 8.2 11.2 5.8 6.1 7.5 11.1 5.74 7.3 6.1 6.5 5.7 4.1 3.8 6.0 3.3 4.2 3.5 7.1 2.9C61487.6 83.9 76.7 85.8 91.1 88.2 82.0 90.0 90.7 87.2 81.8 90.1aValues are in percent. Given are anomalies for 1950–1995 over the entire country (C5095) and for 1900–1995 and 1900–1949over southern Canada (south of 60C176N, S0095 and S0049, respectively). The bottom line shows the sum of the four modes.Figure 1. (opposite) The first four empirical orthogonal functions (EOFs) of the seasonal surface air temperature (SAT)anomalies for 1950–1995 over the entire country (C5095). From top to bottom, the four rows represent the winter (a–c),spring (d–f), summer (g–i) and fall (j–l), respectively. Solid curves denote positive contours, dashed curves, negativecontours, and thick curves, zero contours. The contour interval is 0.02. The EOFs are normalized to unit norm.ACL 8 - 2 WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADAWU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA ACL 8 - 3patterns of individual seasons are very similar to each other,except that mode 2 and mode 3 of the spring data areinterchanged. The first mode is uniform in sign (except thewest coast for the summer) by indicating either warm orcold conditions over the whole domain, while the secondmode displays a southwest-northeast contrast. For eachseason, the first mode can always explain 40–50% of thetotal variance, and the second mode about 20%. The fourleading modes explain altogether about 80% of the totalvariance. The four leading principal components (PCs) (i.e.,EOF time coefficients) are thus used as the inputs to theNLPCA model (Figure 2).[9] The first NLPCA modes of SAT anomalies for fourseasons are shown in Figure 3. For convenience, 3-dimen-sional figures are used, where we can see not only theNLPCA curve in the PC1-PC2-PC33-D space but also itsprojections on PC1-PC2,PC1-PC3and PC2-PC3planes. Inwinter (Figure 3a), we can find a notable curve in the PC1-PC2plane, which indicates considerable nonlinearity, rela-tive to the PCA (straight line). 53.7% of the variance of theoriginal data is explained by the NLPCA mode 1, versus44.6% by the PCA mode 1. The MSE (mean square error)for the NLPCA mode is much smaller than that for the PCAmode with a ratio of 0.865. Here MSE is the mean square ofthe distance between a data point and its projection onto theNLPCA (or PCA) mode, i.e., the unexplained variance. TheMSE ratio can be considered an indicator of the non-linearity: As a ratio of 1 or close to 1 means the NLPCAis essentially linear, while a smaller MSE ratio meansstronger nonlinearity.[10] Figures 3b and 3c show relatively weak nonlinearityin the spring and summer data, with MSE ratios of 0.965and 0.977, respectively, and with slightly higher variancecontributions (Table 2) relative to their PCA mode 1counterparts (Table 1). The nonlinearity is then enhancedduring the fall, as manifested in the increased curvature(Figure 3d). The NLPCA mode 1 for the fall data accountsfor 51.8% of the total variance, versus 40.1% by the PCAmode 1. The MSE ratio is 0.833, indicating a little highernonlinearity than during winter.[11] Unlike a PCA mode, which produces a fixed spatialpattern (the EOF), the NLPCA mode does not give asingle characteristic spatial pattern. For a specific value ofthe NLPC u (see Appendix A), we can use the NN to mapu in the bottleneck neuron onto x0in the output layer(Figure 2) using equations (5) and (6) (in Appendix A).Note that u can be regarded as a simple curvilinearcoordinate system on the NLPCA curve, and thereforeeach value of u corresponds to a pattern within the firstfour PCs. Figure 4 shows the SAT anomaly patternscorresponding to minimum u and maximum u.Wecansee that the spatial patterns on opposite extremes of u arenow asymmetric, i.e., no longer mirror images. Theasymmetry is enhanced as the nonlinearity increases.The spatial patterns associated with minimum u andmaximum u arebasicalysymmetricforthesummer(Figures 4e and 4f), and slightly more asymmetric forthe spring, with the warm center shifted southward relativeto the cooling center in Figures 4c and 4d. The spatialasymmetry gets enhanced in the fall and winter. For thefall, the minimum and maximum u patterns have respec-tively negative and positive anomalies basically coveringthe whole domain. But the cold pattern is centered overthe northeast (Figure 4g), and the warm pattern is centeredtoward the west (Figure 4h). The spatial patterns foropposite values of u are also quite asymmetric duringthe winter. At maximum u, there are positive anomaliesover all of Canada (Figure 4b), similar to the EOF1(Figure 1a), with the warm center over western Canada.At minimum u, there are negative anomalies to the south-west and positive anomalies to the northeast, similar to theEOF2(Figure 1b). It is not surprising because the NLPCAmode 1 combines PC1 and PC2 (Figure 3a).[12] It is worth noting that the winter SAT anomalypattern associated with maximum u (Figure 4b) is similarto the composite results during El Nin˜o years [Hoerling etal., 1997, 2001], while the SAT anomaly pattern associatedwith minimum u (Figure 4a) is different from their compo-site results during La Nin˜a years. Therefore, the NLPCAmode reveals the possible effects from more than ENSO onthe Canadian SAT. This will be further discussed later in thispaper.3.2. S0095[13] Similarly, traditional PCAwas performed on the SATanomalies of S0095 for each season to compress the dataand the first three EOFs are shown in Figure 5. The EOF1and EOF2 are roughly consistent with those of C5095,while the EOF3 patterns are different from those shown inFigure 1. The variance contributions for the four leadingmodes are listed in Table 1. The first four PCs are used asthe inputs to the NLPCA.Figure 2. A schematic diagram of the NN model forcalculating nonlinear PCA (NLPCA). There are three‘‘hidden’’ layers of variables or ‘‘neurons’’ (denoted bycircles) sandwiched between the input layer x on the left andthe output layer x0on the right. Next to the input layer is theencoding layer (with m hidden neurons), followed by the‘‘bottleneck’’ layer (with a single neuron u), which is thenfollowed by the decoding layer. A nonlinear function mapsfrom the higher dimension input space to the lowerdimension bottleneck space, followed by an inverse trans-form mapping from the bottleneck space back to the originalspace represented by the outputs, which are to be as closeto the inputs as possible by minimizing the cost functionJ = hkx C0 x0k2i. Data compression is achieved by thebottleneck, with the bottleneck neuron giving u,thenonlinear principal component. The actual NLPCA usedin this paper has four inputs (and four outputs), has mbetween 2 and 3, and adds a weight penalty term in the costfunction to alleviate overfitting (see Appendix A).ACL 8 - 4 WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA[14] The first NLPCA modes of SAT for four seasons areshown in Figure 6, where we can see similar results asextracted from C5095 although the NLPC curves are notexactly the same. The strongest nonlinearity occurs inwinter with a MSE ratio of 0.879, followed by the fall witha MSE ratio of 0.908. The NLPCA modes explain 63.4%and 60.8% of the total variance of the winter and fall SATanomalies, respectively, versus 58.2% and 55.4% by thePCA modes. Figures 6b and 6c show rather weak non-linearity in the spring and summer data (see also Table 2).[15] In the spatial patterns (Figure 7), for the spring(Figures 7c and 7d) and summer (Figures 7e and 7f), theSAT anomalies on opposite extremes of the NLPC u arebasically similar except for a sign change, resembling theirEOF1 patterns(Figures 5d and 5g). More asymmetry can beseen in the winter (Figures 7a and 7b) and fall (Figures 7gand 7h), and the spatial patterns are in reasonably goodagreement with those shown in Figures 4a and 4b andFigures 4g and 4h, respectively. Although C5095 and S0095are partially overlapping (not completely independent), the–100–50050100–100–50050100–100–50050100pc1(a) DJFpc2pc3–100–50050100–100–50050100–100–50050100pc1(b) MAMpc2pc3–50–2502550–50–2502550–50–2502550pc1(c) JJApc2pc3–50–2502550–50–2502550–50–2502550pc1(d) SONpc2pc3Figure 3. The first NLPCA mode for the seasonal SAT anomalies of C5095 plotted as (overlapping)squares in the PC1-PC2-PC33-D space. The linear (PCA) mode is shown as a dashed line. The NLPCAmode and the PCA mode are also projected onto the PC1-PC2plane, the PC1-PC3plane, and the PC2-PC3plane, where the project NLPCA is indicated by (overlapping) circles, and the PCA by thin solid lines,and the projected data points by scattered dots. Panels (a), (b), (c) and (d) correspond to the winter,spring, summer and fall, respectively.Table 2. Explained Variance by the NLPCA Mode 1 (Ev) and the Ratio (R) Between the MSE of the NLPCA Mode 1and That of the PCA Mode 1aC5095 S0095 S0049DJF MAM JJA SON DJF MAM JJA SON DJF MAM JJA SONEv 53.7 48.4 41.2 51.8 63.4 55.2 48.8 60.8 62.7 56.3 48.3 61.9R 0.865 0.965 0.977 0.833 0.879 0.983 0.986 0.908 0.951 0.960 0.973 0.958aValues are in percent.WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA ACL 8 - 5Figure 4. The SATanomaly patterns (in C176C) of the NLPCA mode 1 extracted from C5095, as the NLPC(i.e., u) of the first NLPCA mode takes its minimum and maximum values. Contour interval is 1C176C. Thefour rows from top to bottom display the winter (a,b), spring (c,d), summer (e,f) and fall (g,h) patterns,respectively.ACL 8 - 6 WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADAFigure 5. Similar to Figure 2, but for S0095. For better visualization, the values have been multipliedby 100 and drawn with contour interval of 2.WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA ACL 8 - 7similarity of the NLPCA results from two data sets suggeststhe significance of the seasonal dependence of the non-linearity of the Canadian SAT, at least for the data after the1950s.3.3. S0049[16] S0049, which is completely independent of C5095, isconsidered in this section. Spatial patterns of the first threePCA modes, shown in Figure 8, are well consistent withthoseextracted fromS0095(Figure5,exceptfortheoppositesign in EOF2 for the winter, spring and summer). Detailedinformationforthe PCA modes arelisted inTable1. Thefirstfour PCs serve as the inputs to the NLPCA.[17] The first NLPCA modes of SAT anomalies for fourseasons are shown in Figure 9. If disregarding the fewoutlier points (e.g., in Figure 9d), no notable nonlinearitycan be seen in any of the four seasons. Table 2 indicated thatthe MSE ratios are all above 0.95 and the explainedvariance percentage differences between nonlinear andlinear modes are also small (not over 2.0%), implying ratherweak nonlinearity. The spatial anomaly patterns associatedwith minimum u and maximum u are generally symmetricfor all four seasons (figures not shown), confirming theweak nonlinearity again.[18] Since S0095 exhibits considerable nonlinearity inwinter and fall, while S0049 does not show apparent non-linearity for all four seasons, nonlinearity for the winter andfall exists mainly in the data after 1950s, suggestinginterdecadal dependence in the nonlinearity of the CanadianSAT.4. Dynamics Related to Canadian SATNLPCA Mode4.1. Winter 500-mb Height[19] As the strong nonlinearity tends to occur in winter,we will examine the winter season more closely. Regres-sions of a winter SAT EOF PC onto the global 500-mbheight anomalies (1950–1995) reveal the spatial patterns ofthe 500-mb heights covarying with the PC, which wasstandardized first. The first winter SAT PC regressed ontothe simultaneous 500-mb height anomalies revealed aPacific-North American (PNA) pattern, where there isgenerally positive height anomalies over Canada, and neg-ative height anomalies over the midlatitude North Pacificand the southeastern United States when the PC is positive(Figure 10a), corresponding to warming over whole Canada(Figure 4b). The second SAT PC regressed onto the winter–100–50050100–100–50050100–100–50050100pc1(a) DJFpc2pc3–100–50050100–100–50050100–100–50050100pc1(b) MAMpc2pc3–50–2502550–50–2502550–50–2502550pc1(c) JJApc2pc3–50–2502550–50–2502550–50–2502550pc1(d) SONpc2pc3Figure 6. Similar to Figure 3, but for S0095.ACL 8 - 8 WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADAFigure 7. Similar to Figure 4, but for S0095.WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA ACL 8 - 9Figure 8. Similar to Figure 5, but for S0049.ACL 8 - 10 WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA500-mb height anomalies gave an NAO pattern, with a largenegative anomaly centered over southwestern Greenlandand northeastern Canada, and a large positive anomalycentered over western Europe and a positive anomaly overthe west coast of Canada when the PC is positive (Figure10b), corresponding to warm western Canada and cooleastern Canada (Figure 4a with signs reversed). Since majornonlinearity occurs between PC1-PC2(Figures 3a and 6a),the leading NLPCA mode thus combines the effects of PNAand NAO on the Canadian winter SAT, suggesting that theeffects of PNA and NAO on Canada may not be totallyindependent. It appears that the SAT PC2tends to take onnegative values when PC1takes on significant negativevalues (Figure 3a), i.e., negative NAO effects, a warmeastern Canada-cool western Canada (Figure 1b with signsreversed), tends to concur with negative PNA effects (coolCanada) (Figure 1a with signs reversed).4.2. The Global SST[20] Similar regressions of a winter SAT PC onto theglobal SST anomalies were performed. The regressions ofthe first winter SAT PC onto SST anomalies yielded aspatial pattern resembling the Pacific Decadal Oscillation(PDO) [Trenberth and Hurrell, 1994]; that is, when the PCis positive (corresponding to warm Canada, Figure 1a),there is cool SSTover the central-western midlatitude NorthPacific and warm waters off the west coast of NorthAmerica (Figure 10c), which is consistent with the ocean’sresponse to the PNA pattern in the atmosphere. The thirdSAT PC regressed on the SST anomalies manifested apattern somewhat similar to ENSO, i.e., warm SST in theeastern-central equatorial Pacific (Figure 10d) when thethird PC is positive, corresponding to warm southernCanada and cool northern Canada (Figure 1c).4.3. Nonlinearity and the NAO[21] We take a closer look at the NLPCA mode for S0095and S0049 in Figure 11 (only projections on the PC1-PC2plane are shown). S0095 and its NLPCA mode are shownby the symbols of plus and overlapping circles, while S0049and its NLPCA mode, by scattered dots and squares,respectively. For comparison, the signs of PC2, also thePC2at the output layer of NLPCA model of S0049 arereversed since the EOF2 of S0049 (Figure 8b) is opposite insign to the EOF2 of S0095 (Figure 5b). We can see thesymbols of ‘‘+’’ and ‘‘.’’ at the same time (1900–1949) are–100–50050100–100–50050100–100–50050100pc1(a) DJFpc2pc3–100–50050100–100–50050100–100–50050100pc1(b) MAMpc2pc3–50–2502550–50–2502550–50–2502550pc1(c) JJApc2pc3–50–2502550–50–2502550–50–2502550pc1(d) SONpc2pc3Figure 9. Similar to Figure 3, but for S0049.WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA ACL 8 - 11close but not exactly overlapping, as the EOFs for two datasets are similar but not identical (Figures 5a–5c and Figures8a–8c). Regarding the NLPCA modes, if we ignore the datapoints of S0095 in the big circle (Figure 11), the two NLPCcurves could be very similar, i.e., it is these points thatbrings the nonlinearity to S0095. These data points are to befound in the winters of 1916, 1933, 1937, 1951, 1955, 1956,1965,1966, 1968, 1969, 1977 and 1982. These years havenegative NAO index values (Figure 12), except for 1933,1937 and 1982, and most of them are within the period of1951–1977. Actually, after a 7-year running mean, theNAO index (thick line in Figure 12) is basically positivefrom 1900 to 1930, normal in the 1930s and 1940s andnegative from 1952 to the early 1970s and then turns to bepositive after the mid 1970s, reaching its positive extreme inthe early 1990s. It is possible that the existence of thenonlinearity in the Canadian winter SAT only after the1950s is due to the strong negative phase of the NAOoccurring from 1950s to the early 1970s. The negative NAOpattern (Figure 10b with signs reversed) and the resultingSAT patterns shown in Figures 4a and 7a are more likely tooccur during the negative phase of the NAO.[22] To check that the downward curve in S0095 (Figure11) is not due to the outliers from a single year, we removedthe data from winter 1969 (which contributed the two pointswith the most negative PC2values) and recomputed theNLPCA solution, which again yielded the downward curvefound in Figure 11.5. Concluding Remarks[23] NLPCA was performed on three subsets of theCanadian seasonal SATs: 1900–1949 and 1900–1995 oversouthern Canada (south of 60C176N), and 1950–1995 over theentire country, with the three data sets named S0049, S0095,and C5095, respectively. The SAT anomalies were found tobe generally linear during 1900–1949 for all seasons inS0049. However, after 1950, the SAT anomalies showedconsiderable nonlinearity in winter (DJF) and fall (SON)(but much weaker nonlinearity in spring and summer),indicating interdecadal and seasonal dependence in thenonlinearity of the Canadian SAT. During winter, theleading NLPCA model reveals asymmetric SAT anomalypatterns: At one extreme of the NLPCA mode, there areFigure 10. The regression coefficients between the winter first PC (of the Canadian SAT anomalies ofC5095) and the global simultaneous (a) 500-mb height anomalies, and (c) SST anomalies; and theregression coefficients (b) between the winter second PC and the 500-mb height anomalies, and (d)between the winter third PC and the SST anomalies. The regression coefficient shown is the slope, withthe PC treated as the independent variable in the regression. Contour interval is 10 in panel (a) and (b),and 0.1 in panel (c) and (d), respectively. The SAT PCs were standardized before performing regression.ACL 8 - 12 WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADAnegative NAO effects with a warm eastern Canada-coolwestern Canada pattern (Figure 4a), and at the otherextreme, positive PNA effects with warming over all ofCanada (Figure 4b). The negative NAO effects also tend toconcur with negative PNA effects, suggesting that the PNAand NAO effects on Canadian SAT may not be independent,and the NLPCA has been successful in combining theeffects of the two.[24] The absence of notable nonlinearity in the SATanomalies during 1900–1949 was puzzling. Since non-–60 –40 –20 0 20 40 60–60–50–40–30–20–10010203040PC1PC2Figure 11. Projections of S0095 and its leading NLPCA mode onto the PC1-PC2plane, shown by thesymbol of plus (+) and overlapping circles. Similarly, projections of S0049 and its leading NLPCA mode,by scattered dots and overlapping squares, respectively. The big circle includes the data points whichpossibly contribute to nonlinearity of SAT anomalies after 1950. The straight line represents the PCAmode 1.Figure 12. Time series of the winter (December through March) index of the North Atlantic Oscillation(NAO), defined as the difference of normalized sea level pressure (SLP) between Lisbon andStykkisholmur (Lisbon minus Stykkisholmur) since 1864 to 2001, shown with the hollow bars. A 7-yearrunning mean of the NAO index is displayed by the thick line.WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA ACL 8 - 13linearity was found in the data during 1900–1995 (S0095),this implied nonlinearity of considerable strength in the dataafter 1950. In fact, the NLPCA mode 1 results from data setS0095 agreed well with those from data set C5095, con-firming the robustness of the nonlinearity found in the SATanomalies in the later period of 1950–1995. Still, there is aconcern that the nonlinearity detected in the 1950–1995period was due to sampling, as nonlinearity was notdetected in the 1900–1949 period. Indeed, with shortclimate records, this possibility cannot be ruled out. How-ever, in the last section, our test with the outlier winter1969data removed from the NLPCA calculation at least elimi-nated the possibility that the detected nonlinearity arosefrom a single outlier year. Furthermore, as negative NAOeffects contributed to the nonlinearity in the winter SAT, andstrong negative NAO occurred during the 1950s to the early1970s but not during the 1900–1949 period, this could be areason for the absence of significant nonlinearity in thewinter SAT data before 1950.[25] The temporal relationship between the NAO andENSO was investigated by Huang et al. [1998] using amulti-resolution cross-spectral technique. Their results showsignificant coherence between NAO and Nin˜o3 SST inabout 70% of the warm events from 1900 to 1995; that is,the strong El Nin˜o events concur with the positive phase ofthe NAO. During relatively weak Nin˜o3 SST anomalies,they found a teleconnection pattern which shows a strongnegative phase of the NAO and a pattern resembling a weakeastward-shifted negative PNA pattern. Interestingly, theirresults generally corroborate the findings in this paper.Appendix A: The NLPCA Method[26] Avariable x, which consists of l spatial stations and nobservations in time, can be expressed in the form x(t)=[x1,...,xl], where xi(i =1,2,..., l), is a time series oflength n. PCA is to find a scalar variable u and an associatedvector a, withuðtÞ¼a C1 xðtÞ; ðA1Þso thathkxðtÞC0auðtÞk2i is minimized; ðA2Þwhere hC1C1C1i denotes a sample or time mean. Here u, calledthe first principal component (PC), is a time series resultingfrom a linear combination of the original variables xi, whilea, the first eigenvector of the data covariance matrix (also-called an empirical orthogonal function, EOF), oftendescribes a spatial pattern. From the residual, x C0 au,thesecond PCA mode can similarly be extracted, and so on forthe higher modes. In practice, the common algorithms forPCA extract all modes simultaneously by calculating theeigenvalues and eigenvectors of the data covariance matrix.[27] The fundamental difference between NLPCA andPCA is that NLPCA allows a nonlinear continuous mappingfrom x to u whereas PCA only allows a linear mapping. Toperform NLPCA, the NN in Figure 2 contains three ‘‘hid-den’’ layers of variables (or ‘‘neurons’’) between the inputand output layers, called the encoding layer, bottleneck, anddecoding layer, respectively.[28] Following Hsieh [2001b], four transfer functions f1,f2, f3, f4are used to map from the input layer to the outputlayer (x ! h(x)! u ! h(u)! x0):hðxÞk¼ f1ððWðxÞx þ bðxÞÞkÞ; ðA3Þu ¼ f2ðwðxÞC1 hðxÞþC22bðxÞÞ; ðA4ÞhðuÞk¼ f3ððwðuÞu þ bðuÞÞkÞ; ðA5Þx0i¼ f4ððWðuÞhðuÞþC22bðuÞÞiÞ; ðA6Þwhere the capital bold font is reserved for matrices and thesmall bold font for vectors, x is the input column vector oflength l,h(x), acolumn vector of length m(mis thenumber ofthe hidden neurons in the encoding layer), W(x)is an m C2 lweight matrix, b(x), a column vector of length m containingthe bias parameters, and k 2 [1, m]. The bottleneck layercontains a single neuron, which represents the nonlinearprincipal component u. The decoding layer contain the samenumber of neurons m as the encoding layer, and the outputlayer is also a column vector of length l.[29] The transfer functions f1and f3are generally non-linear (here taken to be the hyperbolic tangent function),while f2and f4are taken to be the identity function. If thetransfer functions f1and f3are also replaced by a linearfunction, NLPCA essentially reduces to PCA.[30] The cost function J = hkx C0 x0k2i is minimized byfinding the optimal values of W(x), b(x), w(x),C22b(x), w(u),b(u), W(u)andC22bðuÞ. The MSE (mean square error) betweenthe NN output x0and the original data x is thus minimized.[31] Generally,wecanimposetheconstrainthui=0,henceC22bðxÞ¼C0hwðxÞC1 hðxÞi: ðA7ÞThe total number of free (weight and bias) parameters to bedetermined is then 2lm +4m + l. Furthermore, we adopt thenormalization condition that hu2i = 1. This condition isapproximately satisfied by modifying the cost function toJ ¼hkx C0 x0k2iþðhu2iC01Þ2: ðA8Þ[32] The most serious problem with NLPCA is thepresence of local minima in the cost function. As a result,optimizations started from different initial parameters oftenconverge to different minima, rendering the method unsta-ble. This problem can be effectively avoided by adding aweight penalty term into the cost function [Hsieh, 2001b].J ¼hkx C0 x0k2iþðhu2iC01Þ2þ pXkiðWðxÞkiÞ2; ðA9Þwhere p is the weight penalty parameter. With p,theconcavity of the cost function is increased, pushing theweights W(x)to be smaller in magnitude, thereby yieldingsmoother and less nonlinear solutions than when p is smallor zero. With a large enough p, the danger of overfitting isgreatly reduced, hence the optimization can proceed untilconvergence to the global minimum.ACL 8 - 14 WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA[33] The nonlinear optimization was carried out by theMATLAB function ‘‘fminu,’’ a quasi-Newton algorithm.Despite of the weight penalty, there is still no guarantee thatthe optimization algorithm reaches the global minimum.Hence an ensemble of 60 NNs with random initial weightsand bias parameters was run. Also, 20% of the data wasrandomly selected as test data and withheld from the train-ing of the NNs. Runs where the MSE was larger for the testdata set than for the training data set were rejected to avoidoverfitted solutions. Then the NN with the smallest MSEwas selected as the solution. The NLPCA was run repeat-edly with m = 2 and 3, and 30 values of p ranging from 0 to0.18, then the solution with the smallest MSE was chosen asthe desired solution.[34] Acknowledgments. This work was supported by a strategic grantto Hsieh and Shabbar from the Natural Sciences and Engineering ResearchCouncil of Canada, and a contract to Hsieh from the MeteorologicalServices of Canada. We thank Francis Zwiers for helpful comments.ReferencesBunkers, M. J., J. R. Miller, and A. T. DeGaetano, An examination of the ElNin˜o or La Nin˜a related precipitation and temperature anomalies acrossthe northern plains, J. Clim., 9, 147–160, 1996.Hannachi, A., Toward a nonlinear identification of the atmospheric responseto ENSO, J. Clim., 14, 2138–2149, 2001.Hoerling, M. P., A. Kumar, and M. Zhong, El Nin˜o, La Nin˜a and thenonlinearity of their teleconnections, J. Clim., 10, 1769–1786, 1997.Hoerling, M. P., A. Kumar, and T. Xu, Robustness of the nonlinear climateresponses to ENSO’s extreme phases, J. Clim., 14, 1277–1293, 2001.Horel, J. D., and J. M. Wallace, Planetary scale atmospheric phenomenaassociated with the Southern Oscillation, Mon. Weather Rev., 109, 813–929, 1981.Hsieh, W. W., Nonlinear canonical correlation analysis by neural networks,Neural Networks, 13, 1095–1105, 2000.Hsieh, W. W., Nonlinear canonical correlation analysis of the tropicalPacific climate variability using a neural network approach, J. Clim.,14, 2528–2539, 2001a.Hsieh, W. W., Nonlinear principal component analysis by neural networks,Tellus, Ser. A, 53, 599–615, 2001b.Hsieh, W. W., and B. Tang, Applying neural network models to predictionand data analysis in meteorology and oceanography, Bull. Am. Meteorol.Soc., 79, 1855–1870, 1998.Huang, J., K. Higuchi, and A. Shabbar, The relationship between the NorthAtlantic Oscillation and the El Nin˜o-Southern Oscillation, Geophys. Res.Lett., 25, 2707–2710, 1998.Hurrel, J. W., Decadal trends in the North Atlantic Oscillation: Regionaltemperatures and precipitation networks, Sciences, 269, 676–679, 1995.Kalnay, E., et al., The NCEP/NCAR 40-year reanalysis project, Bull. Am.Meteorol. Soc., 77, 437–471, 1996.Kramer, M. 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Khandekar, The impact of El Nin˜o-Southern Oscilla-tion on the temperature field over Canada, Atmos. Ocean, 34, 401–416,1996.Shabbar, A., B. Bonsal, and M. Khandekar, Canadian precipitation patternsassociated with Southern Oscillation, J. Clim., 10, 3016–3027, 1997.Smith, T. M., R. W. Reynolds, R. E. Livezey, and D. C. Stockes, Recon-struction of historical sea surface temperatures using empirical orthogonalfunctions, J. Clim., 9, 1403–1420, 1996.Trenberth, K. E., and J. W. Hurrell, Decadal atmosphere-ocean variations inthe Pacific, Clim. Dyn., 9, 303–319, 1994.Trenberth, K. E., G. W. Branstator, D. Karoly, A. Kumar, N.-C. Lau, andC. Ropelewski, Progress during TOGA in understanding and modellingglobal teleconnections associated with tropical sea surface temperatures,J. Geophys. Res., 103, 14,291–14,324, 1998.Vincent, L. A., and D. Gullet, Canadian historical and homogeneoustemperature datasets for climate change research, Int. J. Clim., 19,1375–1388, 1999.Wallace, J. M., and D. Gutzler, Teleconnection in the geopotential heightfield during the Northern Hemisphere winter, Mon. Weather Rev., 109,784–812, 1981.Zhang, X., L. A. Vincent, W. D. Hogg, and A. Niitsoo, Temperature andprecipitation trends in Canada during the 20th century, Atmos. Ocean, 38,395–429, 2000.C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0W. W. Hsieh and A. Wu, Department of Earth and Ocean Sciences,University of British Columbia, Vancouver, British Columbia, Canada V6T1Z4. (awu@eos.ubc.ca)A. Shabbar, Climate Research Branch, Meteorological Service ofCanada, Downsview, Ontario, Canada M3H 5T4.WU ET AL.: CHARACTERISTICS OF SURFACE AIR TEMPERATURE OVER CANADA ACL 8 - 15


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